Algebraic Analysis Solutions And Exercises Illustrating the Fundamental Theor

George Albert Wentworth

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35a;'^— Ga'^r*^— 9a\ 17. 2a'-a'Z)-a^' + 2Z)l 24. A' ^)2+ 2 a^c^— a' c^- 5^ cl 18. 5'"* + Z)''" — 2. 25. Arn?--ah'' + h''m — m\ 19. Y^"— 23/'"2"— 2?/V"+c'^ 26. I— 6a2 + 27a^ 27. (:i'-3/7 + (l-n^ + y)Cr-y)2;-2l 28. 24m' — 28m'n + 6mn' — TtiI 29. Rr'"+" + :^;"3/'* + a:"*3/"' + 3/"'+^ 30. Rr'* + 207^2/ — o^o(^-{- x^'if' — 2aa;3/^ — y*.
Application' OF the Theory of Divisors. § 24. By Theorem I. We prove that x^ — a"^ is divisible by a; — a always^ x^ — a" is divisible by :^ + a when n is even^ x"^ ...+ a** is divisible by rr + a when 9^ is odd.
By actual division we find in the above cases : ^— !^ r= x''-' + x^-'a + + m"-' + a«-^ (1) 5!+_^ = :r"-^ - x^'-'a + - xa""-' + a""^ (3) Examples.
1. Resolve into factors x^ — y'.
Here x — y is one factor, and by (1) the other is x' + xy + y''.
106 FACTOEING.
2. Resolve a^ + (b — cf.
Here a + (^ — (?) is one factor, and by (3) tlie other is a'-a(b-c) + (h-c)\ 3. Resolve rr^^+1024y^^ This equals (x^f -\-[(2yyY, one factor of which is x^ -{- (2y)^ and by (3) the other factor is (xj - ixy{4:2/) + (xji^yj - x\AyJ + (Aff = x'' - 4:x'y' + IQx'y' - 64.

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